s.t.
Suppose that the righthand side for constraint 1 is increased from 9 to 10.
What does the righthandside range information for constraint 1 tell you about the shadow price for constraint 1?
Question 3 – 16 Marks
Chrystab Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A portfolio consists of U shares of U.S. Oil and H shares of Huber Steel. The annual return for U.S. Oil is $3 per share and the annual return for Huber Steel is $5 per share. U.S. Oil sells for $25 per share and Huber Steel sells for $50 per share. The portfolio has $80,000 to be invested. The portfolio risk index (0.50 per share of U.S. Oil and 0.25 per share for Huber Steel) has a maximum of 700. In addition, the portfolio is limited to a maximum of 1000 shares of U.S. Oil. The linear programming formulation that will maximize the total annual return of the portfolio is as follows:
Max 8U + 5HMaximize total annual return
s.t.
25U + 50H ≤80,000Funds available
0.50U+ .25H ≤ 700 Risk Maximum
1U≤ 1000U.S. Oli Maximum
U, H≤ 0
The sensitivity report for this problem is shown in below in Table 1.
Table 1
Variable Cells 
Model Variable 
Name 
Final Value 
Reduced Cost 
Objective Coefficient 
Allowable Increase 
Allowable Decrease 
U 
U.S Oil 
800.000 
0.000 
3.000 
7.000 
0.5000 
H 
Huber 
1200.000 
0.000 
5.000 
1.000 
3.5000 

Constraints 
Constraint Number 
Name 
Final Value 
Shadow Price 
Constraint R.H. Side 
Allowable Increase 
Allowable Decrease 
1 
Funds available 
80000.000 
0.093 
80000.000 
60000.000 
15000.000 
2 
Risk maximum 
700.000 
1.333 
700.000 
75.000 
300.000 
3 
U.S. Oil maximum 
800.000 
0.000 
1000.000 
1E+30 
200.000 







 What is the optimal solution, and what is the value of the total annual return?
 Which constraints are binding? What is your interpretation of these constraints in terms of the problem?
 What are the shadow prices for the constraints? Interpret each.
 Would it be beneficial to increase the maximum amount invested in U.S. Oil? Why or why not?
Question 48 Marks
The New West Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year’s program. Advertising alternatives include television, radio, and online. Audience estimates, costs, and maximum media usage limitations are as shown:
Constraints 
Television 
Radio 
Online 
Audience per advert 
1000,000 
18000 
40,000 
Cost per Advert 
$2000 
$300 
$600 
Maximummedia usage 
10 
20 
10 
To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized.
 If the promotional budget is limited to $18,200, how many commercial messages should be run on each medium to maximize total audience contact? What is the allocation of the budget among the three media, and what is the total audience reached?
 By how much would audience contact increase if an extra $100 were allocated to the promotional budget?
INDIVIDUAL PROBLEM #6 56 Marks 
Question 1 – 15 – Marks
Consider the following allinteger linear program:
Max1 x_{1+ } 1x_{2 }
s.t.
4×1 + 16x_{2}22
1×1 + 15×2 15
2x_{1+}1x_{2} 9
x_{1}, x_{2} 0 and integer
 Graph the constraints for this problem. Use dots to indicate all feasible integer solutions.
 Solve the LP Relaxation of this problem.
 Find the optimal integer solution
The optimal solution to the LP Relaxation is shown on the above graph to be x1 = 4, x2 = 1.Its value is 5.
The optimal integer solution is the same as the optimal solution to the LP Relaxation.This is always the case whenever all the variables take on integer values in the optimal solution to the LP Relaxation.
Question 2 – 16 Marks
Hawkins Manufacturing Company produces connecting rods for 4 and 6cylinder auto mobile engines using the same production line. The cost required to set up the production line to produce the 4cylinder connecting rods is $2000, and the cost required to set up the production line for the 6cylinder connecting rods is $3500. Manufacturing costs are $15 for each 4cylinder connecting rod and $18 for each 6cylinder connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the following week. If there is a production changeover from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up, the weekly production capacities are 6000 6cylinder connecting rods and 8000 4cylinder connecting rods. Let
x_{4} = the number of 4cylinder connecting rods produced next week
x_{6}= the number of 6cylinder connecting rods produced next week.
S_{4} = 1 if the production line is set up to produce the 4cylinder connecting rods; 0 if
otherwise
S_{6} = 1 if the production line is set up to produce the 6cylinder connecting rods; 0 if
otherwise
 Using the decision variables x_{4} and s_{4}, write a constraint that limits next week’s pro duction of the 4cylinder connecting rods to either 0 or 8000 units.
 Using the decision variables X_{6} and S_{6}, write a constraint that limits next week’s pro duction of the 6cylinder connecting rods to either 0 or 6000 units.
 Write three constraints that, taken together, limit the production of connecting rods for next week.
 Write an objective function for minimizing the cost of production for next week.
Question 3 – 15 – Marks
Consider again the Ohio Trust Inc. problem described in Problem 15. Suppose only a limited number of PPBs can be placed. Ohio Trust would like to place this limited number of PPBs in counties so that the allowable branches can reach the maximum possible population. The file Ohio Trust Pop contains the county adjacency matrix described in Problem 15 as well as the population of each county.
 Assume that only a fixed number of PPBs, denoted k. can be established. Formulate a linear binary integer program that will tell Ohio Trust Inc. where to locate the fixed number of PPBs in order to maximize the population reached.
 Suppose that two PPBs can be established. Where should they be located to maximize the population served?
 Solve your model from part a for allowable number of PPBs ranging from 1 to 10. In other words, solve the model 10 times, k set to 1,2, . . . , 10. Record the population reached for each value of k. Graph the results of this analysis by plotting the population reached versus number of PPBs allowed. Based on their cost calculations, Ohio Trust considers an additional PPB to be fiscally prudent only if it increases the population reached by at least 500,000 people. Based on this graph, what is the number of PPBs you recommend to be implemented?
Hint:
Introduce variable yi = 1 if it is possible to establish a branch in county i, and
yi =0 otherwise; that is, if county i is covered by a PPB, then the population can be counted as covered.
Question 4 – 10 Marks
The employee credit union at State university is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in riskfree securities to stabilize income. The various revenue producing investments together with annual rates of return are as follows:
Types of Load/Investment 
Annual Rate of Return (%) 
Automobile Loans 
8 
Furniture loans 
10 
Other secured loans 
11 
Signature loans 
12 
Riskfree securities 
9 
The credit union will have $2 million available for investment during the coming year. State laws and credit union policies impose the following restrictions on the composition of the loans and investments:
 Riskfree securities may not exceed 30% of the total funds available for investment.
 Signature loans may not exceed 10% of the funds invested in all loans (automobile, furniture, other secured, and signature loans).
 Furniture loans plus other secured loans may not exceed the automobile loans.
 Other secured loans plus signature loans may not exceed the funds invested in riskfree securities
 How should the $2 million be allocated to each of the loan/investment alternatives to maximize total annual return?
 What is the projected total annual return?